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2011 | 9 | 4 | 874-887
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On a generalized Stokes problem

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EN
Abstrakty
EN
We deal with a generalization of the Stokes system. Instead of the Laplace operator, we consider a general elliptic operator and a pressure gradient with small perturbations. We investigate the existence and uniqueness of a solution as well its regularity properties. Two types of regularity are provided. Aside from the classical Hilbert regularity, we also prove the Hölder regularity for coefficients in VMO space.
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Bibliografia
  • [1] Adams R.A., Fournier J.J.F., Sobolev Spaces, 2nd ed., Pure Appl. Math. (Amst.), 140, Elsevier/Academic Press, Amsterdam, 2003
  • [2] Bulíček M., Málek J., Rajagopal K.R., Navier’s slip and evolutionary Navier-Stokes-like systems with pressure and shear-rate dependent viscosity, Indiana Univ. Math. J., 2007, 56(1), 51–85 http://dx.doi.org/10.1512/iumj.2007.56.2997
  • [3] Chiarenza F., L p regularity for systems of PDEs, with coeficients in VMO, In: Nonlinear Analysis, Function Spaces and Applications, 5, Proceedings of the 5th Spring School, Prague, May 23–28, 1994, Prometheus, Prague, 1994, 1–32
  • [4] Daněček J., John O., Stará J., Morrey space regularity for weak solutions of Stokes systems with VMO coefficients, Ann. Mat. Pura Appl. (in press), DOI: 10.1007/s10231-010-0169-7
  • [5] Franta M., Málek J., Rajagopal K.R., On steady flows of fluids with pressure- and shear-dependent viscosities, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 2005, 461(2055), 651–670 http://dx.doi.org/10.1098/rspa.2004.1360
  • [6] Friedman A., Partial Differential Equations, Holt, Rinehart and Winston, New York-Montreal-London, 1969
  • [7] Giaquinta M., Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Ann. of Math. Stud., 105, Princeton University Press, Princeton, 1983
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  • [9] Huy N.D., On Existence and Regularity of Solutions to Perturbed Systems of Stokes Type, Ph.D. thesis, Charles University, Prague, 2006
  • [10] Huy N.D., Stará J., On existence and regularity of solutions to a class of generalized stationary Stokes problem, Comment. Math. Univ. Carolin., 2006, 47(2), 241–264
  • [11] Kufner A., John O., Fučík S., Function Spaces, Monographs Textbooks Mech. Solids Fluids: Mech. Anal., Noordhoff, Leyden, 1977
  • [12] Lax P.D., Functional Analysis, Pure Appl. Math. (N. Y.), John Wiley & Sons, New York, 2002
  • [13] Málek J., Mingione G., Stará J., Fluids with pressure dependent viscosity: partial regularity of steady flows, In: EQUADIFF 2003, Proceedings of the International Conference of Differential Equations, Hasselt/Diepenbeek, July 22–26, 2003, World Scientific, Hackensack, 2005, 380–385 http://dx.doi.org/10.1142/9789812702067_0059
  • [14] Nečas J., Hlaváček I., Mathematical Theory of Elastic and Elasto-Plastic Bodies: an Introduction, Stud. Appl. Mech., 3, Elsevier, Amsterdam-New York, 1980
  • [15] Renardy M., Some remarks on the Navier-Stokes equations with a pressure-dependent viscosity, Comm. Partial Differential Equations, 1986, 11(7), 779–793 http://dx.doi.org/10.1080/03605308608820445
  • [16] Rudin W., Functional Analysis, McGraw-Hill Series in Higher Mathematics, McGraw-Hill, New York-Düsseldorf-Johannesburg, 1973
  • [17] Schechter M., Principles of Functional Analysis, Academic Press, New York-London, 1971
  • [18] Sohr H., The Navier-Stokes Equations, Birkhäuser Adv. Texts Basler Lehrbucher, Birkhäuser, Basel, 2001
  • [19] Troianiello G.M., Elliptic Differential Equations and Obstacle Problems, Univ. Ser. Math., Plenum Press, New York, 1987
Typ dokumentu
Bibliografia
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bwmeta1.element.doi-10_2478_s11533-011-0047-6
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